## Flow Rate

08 May

I received an extra credit essay from one of my students based on a question from the textbook that I had to do a little modeling to understand. The question was one about patients with atherosclerosis that could be explained using Poiseuille’s Law. This Law describes the relationship between the flow rate, pressure, radius and viscosity of a liquid flowing through a vessel.

Basically, it is presented as:

Flow Rate = change in Pressure * pi * radius^4* Length of the vessel * viscosity

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The question asks, ‘why symptoms of myocardial ischemia do not usually occur until ~75% of a vessel has been occluded.’

The easy answer is that that is the cutoff after which the amount of blood required to provide Oxygen sufficient for the heart’s metabolism is insufficient. However, this can be visualized qualitatively simply by graphing the equation. To do this, I made up a quick spreadsheet and just plugged in ‘1’ for all the variables, then solved for the flow rate. From here, I simply plugged in fractions into the radius variable.

Here’s the raw data: 1.00 – 0.75 (i.e. a 75% blockage) = 0.25 is the number from the question. Here’s the analysis: Note how the Flow Rate has dropped to essentially ZERO when the radius is occluded 75%.

There may be more to this, but I think that just looking at this analysis of the equation answers a lot.

ps – I just spent a hell of a lot of time and effort messing around in the terminal of my mac changing the screen capture file type all to realize that it wasn’t my mac that was the problem at all – I simply was not using the largest image type available in wordpress and then tried to scale up my image after it was inserted – don’t do this. You lose all of your image quality.